The Calabi-type conjecture for generalized Kähler structures

Let $(M^{2n},g,J_\pm)$ be a generalized Kähler structure satisfying $[J_+,J_-]=0$. Let $\mathcal H$ be the space of formally generalized Kähler classes, and let $P$ be the associated elliptic operator. Given $\phi\in[P]\in\mathcal H$, Calabi-type conjecture. There exists a unique $\omega_f\in[\omega]$ such that

$$P(\omega_f)=\phi.$$

In the commuting case this discussion reduces to the classical Kähler setting, where the conjecture reduces to the Calabi conjecture and is therefore solved by Yau. The generalized statement is presented as an elliptic analogue of pluriclosed flow.